For a resting uniform continuum gas with pair interactions depending on relative positions, shifting every particle momentum by leaves interaction energy unchanged and changes the microscopic kinetic energy by . The resting ground state has total momentum . The resulting boost energy uses the bare mass even in an interacting Fermi liquid; matching this with the Landau quasiparticle energy functional constrains its interaction kernel.
With parallel inertial axes and absolute time, differentiating a Galilean transformation adds relative velocities as ordinary vectors. This is the low-speed limit of relativistic velocity addition. At speeds comparable with the speed of light, absolute time is no longer the appropriate frame transformation, and the relativistic denominator cannot be neglected.
The laws of Newtonian mechanics have the same form in every inertial frame related by a Galilean transformation.
A dynamical symmetry under Galilean transformations. For an isolated pair with constant masses, a force law depending only on relative positions and relative velocities has this invariance. Equal-and-opposite forces alone do not imply it: their magnitude can depend on centre of mass velocity relative to a preferred frame.
For a Galilean transformation , , with constant rotation and boost , velocity transforms as and acceleration as . With invariant inertial mass, Newton's second law retains its form only if the actual interaction laws and their physical background data transform to give . Acceleration invariance alone does not prove covariance of an arbitrary force law. For example, resistance depends on velocity relative to a medium, whose velocity must transform too.
